Long-Range Correlations in Elastic Moduli and Local Stresses at the Unjamming Transition

TL;DR

This study investigates long-range correlations in local elastic moduli and stresses in jammed solids, revealing universal stress correlation behavior near unjamming and potential anisotropic shear modulus correlations.

Contribution

It uncovers the universal quadratic pressure dependence of stress correlations and the anisotropic, long-range correlations of local shear moduli in jammed materials.

Findings

Stress correlations scale as p^2 near unjamming.

Local shear moduli exhibit anisotropic, long-range correlations.

Total modulus shows long-range correlations, unlike affine responses.

Abstract

We explore the behavior of spatially heterogeneous elastic moduli as well as the correlations between local moduli in model solids with short-range repulsive potentials. We show through numerical simulations that local elastic moduli exhibit long-range correlations, similar to correlations in the local stresses. Specifically, the correlations in local shear moduli exhibit anisotropic behavior at large lengthscales characterized by pinch-point singularities in Fourier space, displaying a structural pattern akin to shear stress correlations. Focussing on two-dimensional jammed solids approaching the unjamming transition, we show that stress correlations exhibit universal properties, characterized by a quadratic $p^2$ dependence of the correlations as the pressure $p$ approaches zero, independent of the details of the model. In contrast, the modulus correlations exhibit a power-law dependence with different exponents depending on the specific interaction potential. Furthermore, we illustrate that while affine responses lack long-range correlations, the total modulus, which encompasses non-affine behavior, exhibits long-range correlations.